A flat connection on a principal fiber bundle over M with structure group G defines a homomorphism from the fundamental group of M to G via parallel transport along closed curves. (Otherwise put, the holonomy group of a flat connection is a homomorphic image of the fundamental group.) This defines a reduction of structure group to this so called monodromy group. In particular, if M is simply connected, then the bundle is trivial. Hence the Hopf fibration does not admit a flat connection. (This can also be seen from Chern-Weil theory, which computes characteristic classes from curvature expressions.)

If G(n) is a classical series of simple Lie groups, then a natural action of G(k) on G(n), k

Dear  All,

what you reported is correct, but it is only a part of the full-story. In fact, we can always find flat local (k+1)-connections on some fiber bundle in the category of differential manifolds. This of course holds also for all Hopf fibrations.

(For details please look to the following my paper: DOI: 10.1023/A:1006346916360.)

But isn't obvious that any trivial bundle admits an obvious flat connection? If so, then it sufficies to observe that all fiber bundles are, by definition, locally trivial.

Yes, but any local section identifies a non-trivial flat local (k+1)-connection.

Dear Ali,

I agree to Andreas Cap. A similar question would be: Does every compact surface  S allows for a flat metric? No since by Gauss-Bonnet,  \int_S  K dv = 2\pi \chi(S)  where \chi(S) is the Euler number of S (number of faces - number of edges + number of vertices for a triangulation of S). This is a global (topological) invariant which does not depend on the particular Riemannian metric on S. Similar for principal bundles over compact manifolds M. A flat connection corresponds to a flat metric in the toy example. There is no local restriction for the curvature tensor of a connection on a principal bundle (like there is no local obstruction against a flat metric on a surface - you always may have flat patches), but there are global topological obstructions: the characteristic numbers of the bundle (Pontrjagin and Chern numbers) arising by integrating over M certain homogenous polynomials in the components of the curvature tensor of the connection, polynomials which are invariant under coordinate transformations. Eg. if R_{ij})_{ab} are the components of the curvature tensor, then  trace ((R_{ij})^k)  is such a polynomial. If the connection is flat, all (R_{ij}_{ab} = 0, hence these integrals are zero. But they are topological invariants of the bundle which are nonzero in most cases, which obstructs the existence of flat connections.

Best regards

Jost

S. Kobayashi and K.Nomizu, Foundations of differential geometry, Vol. 1, Chap. II, 9, Chap. V, 4,  and Vol.2, Chap. XII 

The answer is obviously no since a flat connection implies triviality and there are many examples of non-trivial principle bundles.

Dear Maurice,

I agree, but to be precise: flatness implies triviality only if the base space is simply connected. When the universal cover of the base is compact, we may use this argument.

Best regards

Jost